A sparse image method for BEM capacitance extraction

Abstract

Boundary element methods (BEM) are often used for complex 3-D capacitance extraction because of their efficiency, ease of data preparation, and automatic handling of open regions. BEM capacitance extraction, however, yields a dense set of linear equations that makes solving via direct matrix methods such as Gaussian elimination prohibitive for large problem sizes. Although iterative, multipole-accelerated techniques have produced dramatic improvements in BEM capacitance extraction, accurate sparse approximations of the electrostatic potential matrix are still desirable for the following reasons. First, the corresponding capacitance models are sufficient for a large number of analysis and design applications. Moreover, even when the utmost accuracy is required, sparse approximations can be used to precondition iterative solution methods.In this paper, we propose a definition of electrostatic potential that can be used to formulate sparse approximations of the electrostatic potential matrix in both uniform and multilayered planar dielectrics. Any degree of sparsity can be obtained, and unlike conventional techniques which discard the smallest matrix terms, these approximations are provably positive definite for the troublesome cases with a uniform dielectric and without a groundplane.

Bounds For BEM Capacitance Extraction

“…As we will show, the shift-truncate potential method proposed in [1] and depicted in Fig. 3 yields such bounds.…”

“…Several different techniques for the localization of the extracted capacitance have been developed, such as scanning methods [4] [7] which build capacitance databases of the elementary conductor constellations encountered in the circuit, and windowing techniques. This paper proves that the shift-truncate method from [1] and the windowing method yield opposite bounds for the exact values of the mutual and self capacitances. We also introduce the metal shell [2] featuring the same bounds as shift-truncate and the homogenously filled shell which provides a higher accuracy than shift-truncate [2] in many cases.…”

“…In this section we will show that the windowing method yields lower bounds for the two-terminal self capacitances and upper bounds for the two-terminal mutual capacitances, and that the corresponding bounds for the shift-truncation technique [1] are the opposite. Background information about BEM and the terminology used here can be found in section 2 of [1].…”

“…For this reason it is imperative that the shift-truncate scheme must produce a positive definite potential matrix. This can be assured by using shells that are physically based [1].…”

“…Since is positive definite [1], and the solution of (10) exists and is unique. Equation (10) can be also written as (11) and its solution must exist and be unique (equivalence with (10)).…”